I had this question on an Algorithms test yesterday, and I can\'t figure out the answer. It is driving me absolutely crazy, because it was worth about 40 points. I figure
This is not a solution, but a similar line of thought to what Olexiy was thinking
I was playing around with creating sequences with maximum number of ones, and they are all quite interesting, I got up to 125 digits and here are the first 3 numbers it found by attempting to insert as many '1' bits as possible:
Notice they are all fractals (not too surprising given the constraints). There may be something in thinking backwards, perhaps if the string is not a fractal of with a characteristic, then it must have a repeating pattern?
Thanks to beta for the better term to describe these numbers.
Update: Alas it looks like the pattern breaks down when starting with a large enough initial string, such as: 10000000000001:
100000000000011
10000000000001101
100000000000011011
10000000000001101100001
100000000000011011000011
10000000000001101100001101
100000000000011011000011010000000001
100000000000011011000011010000000001001
1000000000000110110000110100000000010011
1000000000000110110000110100000000010011001
10000000000001101100001101000000000100110010000000001
10000000000001101100001101000000000100110010000000001000001
1000000000000110110000110100000000010011001000000000100000100000000000001
10000000000001101100001101000000000100110010000000001000001000000000000011
1000000000000110110000110100000000010011001000000000100000100000000000001101
100000000000011011000011010000000001001100100000000010000010000000000000110100001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001
100000000000011011000011010000000001001100100000000010000010000000000000110100001001000001
1000000000000110110000110100000000010011001000000000100000100000000000001101000010010000010000001
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